Inverse-producing extensions of Topological Algebras/topological algebra

Definition: Topological Vector Space
A topological vector space $V$ over $$\mathbb{K}$$ is a vector space over the body $$\mathbb{K}$$ that has a topology with which scalar multiplication and addition are continuous mappings.



\begin{array}{rcl} \cdot : \mathbb{K} \times V &\longrightarrow & V \quad (\lambda,v) \longmapsto \lambda \cdot v \\ + : V \times V &\longrightarrow & V \quad (v,w) \longmapsto v+w \end{array} $$

In the following, for all topological vector spaces, we shall use the Hausdorff property be assumed.

Definition: Neighbourhood
Let $$(X,\mathcal{T})$$ be a topological space with a topology $$\mathcal{T}$$ as a system of open sets $$\mathcal{T}\subset \wp(X)$$ and $$a\in X$$, then denote


 * $$\mathfrak{U}_{\mathcal{T}} (a) := \left\{ U \subseteq X \, : \, \exists_{U_o \in \mathcal{T}} :\, a \in U_o \subseteq U \right\}$$ the set of all neighbourhoods from the point $$a$$,
 * $$\stackrel{o}{\mathfrak{U}}_{\mathcal{T}} (a):= \mathfrak{U}_{\mathcal{T}} (a) \cap \mathcal{T}$$ the set of all open Neighbourhoods from the point $$a$$,
 * $$\overline{\mathfrak{U}}_{\mathcal{T}} (a) := \left\{ \overline{U} \, : \, U \in \mathfrak{U}_{\mathcal{T}} (a) \right\}$$ the set of all closed neighbourhoods of point $$a$$.

Remark: Indexing with topology
If no misunderstanding about the underlying topological space can occur, the index $$\mathcal{T}$$ is not included as a designation of the topology used.

Remark: Analogy to the epsilon neighbourhood
In convergence statements in the real numbers one usually considers only $$\varepsilon$$ neighbourhood. In doing so, one would actually have to consider in topological spaces for arbitrary neighbourhoods from $$U \in \mathfrak{U}_{\mathcal{T}} (a)$$ find an index bound $$i_U\in I$$ of a net $$(x_i)_{i\in I}$$ above which all $$x_i \in U$$ lie with $$i \geq i_U$$. However, since the $$\varepsilon$$ neighbourhoods are an neighbourhood basis, by the convergence definition one only needs to show the property for all neighbourhoods with $$\varepsilon > 0$$.

Convergence in topological spaces
Let $$(X,\mathcal{T})$$ be a topological space, $$a\in X$$, $$I$$ an index set (partial order) and $$(x_{i})_{i\in I}\in X^{I}$$ a mesh. The convergence of $$(x_{i})_{i\in I}\in X^{I}$$ against $$a\in X$$ is then defined as follows:

\begin{array}{rcl} \displaystyle{\stackrel{\mathcal{T}}{\lim_{i\in I}} \, x_i = a } & :\longleftrightarrow & \forall_{U \in \mathfrak{U}_{\mathcal{T}} (a)} \exists_{i_U \in I}                 \forall_{i \geq i_U} : \, x_i \in U \\ \end{array} $$. (where "$$\leq$$" for $$I$$ is the partial order on the index set).

Definiton: Neighbourhood basis
Let $$(X,\mathcal{T})$$ be a topological space, $$a\in X$$ and $$\mathfrak{U}_{\mathcal{T}} (a) $$ the set of all neighbourhoods of $$a\in X$$. $$\mathfrak{B}_{\mathcal{T}} (a) $$ is called the neighbourhood basis of $$\mathfrak{U}_{\mathcal{T}} (a) $$ if for every :$$\mathfrak{B}_{\mathcal{T}} (a) \subseteq \mathfrak{U}_{\mathcal{T}} (a) \wedge \forall_{U\in \mathfrak{U}_{\mathcal{T}} (a)} \exists_{B \in \mathfrak{B}_{\mathcal{T}} (a)}: \, B \subseteq U$$.

Remark: Epsilon spheres in normalized spaces
Let $$(V,\| \cdot \|)$$ be a normed space, then the $$\varepsilon$$ spheres form
 * $$B_\varepsilon^{\| \cdot \|}(a) := \left\{v\in V\, ; \, \|v-a\| < \varepsilon \right\}$$

an ambient basis of $$\mathfrak{B}_{\mathcal{T}} (a) $$ the set of all environments of $$\mathfrak{U}_{\mathcal{T}} (a) $$ of $$a\in V$$.

Learning Task 1
Let $$(X,\mathcal{T})$$ be a toplogic space with chaotic topology $$\mathcal{T}:= \{\emptyset, X\}$$.
 * Determine $$\mathfrak{U}_{\mathcal{T}} (a) $$ for any $$a \in X$$.
 * Show that any sequence $$(x_{n})_{n\in \mathbb{N}}\in X^{\mathbb{N}}$$ converges in $$(X,\mathcal{T})$$ against any limit $$a \in X$$.

Learning Task 2
Let $$(X,d)$$ be a metric space with the discrete topology given by the metric:
 * $$d(x,y):=

\left\{\begin{array}{lcl} 0 & \mbox{ for } & x = y \\ 1 & \mbox{ for } & x \not= y \\ \end{array}\right.$$.
 * Determine $$\mathfrak{U}_{\mathcal{T}} (a) $$ for any $$a \in X$$.
 * How many sets make up $$\mathfrak{B}_{\mathcal{T}} (a) $$ minimal for any $$a \in X$$?
 * Formally state all sequences $$(x_{n})_{n\in \mathbb{N}}\in X^{\mathbb{N}}$$ in $$(X,d)$$ that converge to a limit $$a \in X$$!

Definition: open sets
Let $$(X,\mathcal{T})$$ be a topological space and $$\mathcal{T}\subseteq \wp(X)$$ be the system of open sets, that is:
 * $$ U\subseteq X \mbox{ open } :\Longleftrightarrow U \in \mathcal{T}$$.

Task
Let $$(\mathbb{R},\mathcal{T})$$ be a topological space on the basic set of real numbers. However, the topology does not correspond to the Euclic topology over the set $$|\cdot |$$, but the open sets are defined as follows.
 * $$ U \in \mathcal{T} \mbox{ open } :\Longleftrightarrow U=\emptyset \mbox{ or } U\subseteq \mathbb{R} \, \mbox{ with }  \, U^c \mbox{ countable}$$

Here $$ U^c := \mathbb{R}\setminus U$$ is the complement of $$U$$ in $$\mathbb{R}$$.
 * Show that $$(\mathbb{R},\mathcal{T})$$ is a topological space.
 * Show that the sequence $$\left( \frac{1}{n} \right)_{n\in\mathbb{N}}$$ does not converge to $$0$$ in the topological space $$(\mathbb{R},\mathcal{T})$$.

Remark: open - closed
By the system of open sets in a topology $$\mathcal{T}\subseteq \wp(X)$$ the closed sets of the topology are also defined at the same time as their complements.

Definition: closed sets
Let $$(X,\mathcal{T})$$ be a topological space and $$\mathcal{T}\subseteq \wp (X)$$ be the system of open sets.
 * $$ M \subseteq X \mbox{ completed } :\Longleftrightarrow \exists_{U \in \mathcal{T}} : \, M = U^c := X\setminus U$$

Definition: open kernel
Let $$(V,\mathcal{T})$$ be a topological space and $$M\subset V$$, then the open kernel $$\stackrel{\circ}{M}$$ of $$M$$ is the union of all open subsets of $$M$$.
 * $$ \stackrel{\circ}{M} := \bigcup_{U \in \mathcal{T}, U\subseteq M} U $$.

Definition: closed hull
Let $$(X,\mathcal{T})$$ be a topological space. The closed hull $$\overline{M}$$ of $$M$$ is the intersection over all closed subsets of $$W= U^c$$ containing $$M$$ and $$U$$ is open.
 * $$ \overline{M} := \bigcap_{U \in \mathcal{T}, U^c := X\setminus U \supseteq M} U^c $$

Definition: edge of a set
The topological edge $$\partial M$$ of $$M$$ is defined as follows:
 * $$\partial M := \overline{M} \backslash \stackrel{\circ}{M}$$

Remark: sequences and nets
In metric spaces, one can still work with the natural numbers as countable index sets. In arbitrary topological spaces one has to generalize the notion of sequences to the notion of nets.

Definition: nets
Let $$T$$ be a topological space and $$I$$ an index set (with partial order), then $$T^I$$ denotes the set of all families indexed by $$I$$ in $$T$$:

T^I :=\{(t_i)_{i\in I}:t_i\in T \mbox{ for all }i\in I\} $$

Definition: finite sequences
Let $$V$$ be a vector space, then $$c_{oo}(V)$$ denotes the set of all finite sequences with elements in $$V$$:

c_{oo}(V):=\left\{ (v_n)_{n\in \mathbb{N}_0} \in V^{\mathbb{N}_0}: \exists_{\displaystyle N\in \mathbb{N}_0}  \forall_{\displaystyle n\geq N} : \, v_n=0 \right\}. $$

Definition: Algebra
An algebra $$A$$ over the body $$\mathbb{K}$$ is a vector space over $$\mathbb{K}$$ in which a multiplication is an inner join

\cdot : A \times A \longrightarrow A \quad (v,w) \longmapsto v\cdot w $$ is defined where for all $$x,y,z\in A$$ and $$\lambda \in \mathbb{K}$$ the following properties are satisfied:

\begin{array}{rcl} x\cdot (y\cdot z) &=& (x\cdot y)\cdot z\\. x\cdot (y+z) &=& x\cdot y + x \cdot z \\ (x+y)\cdot z &=& x\cdot z + y\cdot z \\ \lambda \cdot (x\cdot y) &=& (\lambda \cdot x)\cdot y = x\cdot (\lambda \cdot y)   \end{array} $$

Definition: topological algebra
A topological algebra $$(A,\mathcal{T}_A)$$ over the body $$\mathbb{K}$$ is a topological vector space $$(A,\mathcal{T}_A)$$ over $$\mathbb{K}$$, where also multiplication is

\cdot : A \times A \longrightarrow A \quad (v,w) \longmapsto v\cdot w $$ is a continuous inner knotting.

Continuity of multiplication
Continuity of multiplication means here:

\forall_{\displaystyle U\in \mathfrak{U}\, (0)}  \exists_{\displaystyle V\in \mathfrak{U}\, (0)} : V\cdot V = V^2 \subset U $$

Multiplicative topology - continuity
The topology is called multiplicative if holds:

\forall_{\displaystyle U\in \mathfrak{U}\, (0)}  \exists_{\displaystyle V\in \mathfrak{U}\, (0)} : V^2 \subset V \subset U  $$

Remark: Multiplicative topology - Gaugefunctionals
In describing topology, the Topologization_lemma for algebras shows that the topology can also be described by a system of Gaugefunctionals

Unitary algebra
The algebra $$A$$ is called unital if it has a neutral element $$e$$ of multiplication. In particular, one defines $$x^o:=e$$ for all $$x\in A$$. The set of all invertible (regular) elements is denoted by $${\mathcal{G}} (A)$$. Non-invertible elements are called singular.

Task: matrix algebras
Consider the set $$V$$ of square $$2\times 2$$ matrices with matrix multiplication and the maxmum norm of the components of the matrix. Try to prove individual properties of an algebra ($$V$$ is a non-commutative unitary algebra). For the proof that $$V$$ with matrix multiplication is also a topological algebra, see Topologization Lemma for Algebras.

Definition: sets and links
Let $$(A,\mathcal{T})$$ be a topological algebra over the body $$\mathbb{K}$$, $$\Lambda\subset \mathbb{K}$$ and $$M_1,M_2$$ be subsets of $$A$$, then define

\begin{array}{rcl} M_1 \times M_2 &:=& \{ (m_1,m_2) \in A \times A \,:\, m_1\in M_1 \wedge m_2 \in M_2\} \\ M_1+ M_2 &:=& \{ m_1+m_2 \,:\, m_1\in M_1 \wedge m_2 \in M_2\} \\ M_1\cdot M_2 &:=& \{m_1\cdot m_2 \,:\, m_1\in M_1 \wedge m_2 \in M_2\}\\ \Lambda \cdot M_1 &.=& \{\lambda\cdot m_1 \,:\, m_1\in M_1 \wedge \lambda \in \Lambda\}. \\   \end{array} $$

Learning Tasks
Draw the following set $$M_k$$ of vectors as sets of points in the Cartesian coordinate system $$\mathbb{R}^2$$ with $$M_1 := \left\{ \begin{pmatrix} 1\ 2\end{pmatrix}, \begin{pmatrix} 1\ 0 \end{pmatrix} \right\}$$ and $$M_2 := \left\{ \begin{pmatrix} 3\ 2\end{pmatrix}, \begin{pmatrix} 0\ 1 \end{pmatrix} \right\}$$ and the following intervals $$[a,b] \in \mathbb{R}$$:
 * $$[1,4] \times [2,3]$$.
 * $$ M_1+ M_2$$.
 * $$ [1,2] \cdot M_1$$.

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